Question

If $f\left(x\right)={\int }_0^{x}t(\sin x-\sin t)dt$ then $f^{\prime }\left(x\right)$?

Answer 1

Given,

$f\left(x\right)={\int }_0^{x}t(\sin x-\sin t)dt;$

Now applying Libnitz's rule fon differentiating under integration we get,

$f^{\prime }\left(x\right)=x(\sin x-\sin x).1-0+{\int }_0^{x}t\cos xdt$

on, $f^{\prime }\left(x\right)=\left(\cos x\right)\left(x^2/2\right)$